`Abstract and Applied AnalysisVolumeΒ 2008, Article IDΒ 135237, 10 pageshttp://dx.doi.org/10.1155/2008/135237`
Research Article

A Functional Equation Originating from Elliptic Curves

1National Institute for Mathematical Sciences, 385-16 Doryong-Dong, Yuseong-Gu, 305-340 Daejeon, South Korea
2College of Liberal Arts, Kyung Hee University, 449-701 Yongin, South Korea

Received 17 November 2007; Revised 1 February 2008; Accepted 5 April 2008

Academic Editor: JohnΒ Rassias

Copyright Β© 2008 Won-Gil Park and Jae-Hyeong Bae. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We obtain the general solution and the stability of the functional equation . The function having level curves as elliptic curves is a solution of the above functional equation.

1. Introduction

A graph of an equation of the form is called an elliptic curve [1], where and are constants. Since has level curves as elliptic curves, functional equations having the mapping as a solution are helpful to study cryptography and their applications.

Recently, Jun and Kim [2] solved the cubic functional equation

We consider the quadratic functional equationand the -dimensional vector variable functional equation

The function given by is a particular solution of (1.1), and the function given by is a particular solution of (1.2). The function given by is a particular solution of (1.3). The functional (1.3) is a mixed type of (1.1) and (1.2).

Problems on solutions or stability of various functional equations have been extensively investigated by a number of mathematicians [3β12].

In this paper, let , and be real vector spaces. We find out the general solution and investigate the stability of (1.2) and (1.3).

2. Solutions

We find out the general solution of the functional (1.2) as follows.

Lemma 2.1. A mapping satisfies (1.2) if and only if there exists a symmetric biadditive mapping such thatfor all .

Proof. We first assume that is a solution of (1.2). Let be a mapping given by for all . Then, andfor all . Putting in (2.2), is even. Replacing by in (2.2),for all . Taking in (2.2), we get thatfor all . Interchanging and in (2.4), we see thatfor all . By (2.3), (2.4), and (2.5), we obtain that for all . Letting in (2.6), we get thatfor all . By (2.6) and (2.7), we obtain thatfor all . By [13], there exists a symmetric biadditive mapping such that for all .
Conversely, we assume that there exists a symmetric biadditive mapping such that for all . Since is biadditive,for all .

Example 2.2. Let be a rational number, where and are integers with . Define and . Let be the set of rational numbers and let be an elliptic curve over . The addition in is given in [1]. The height function is defined by for all . The canonical height function given byis a solution of (1.2).

We find out the general solution of the functional (1.3) as follows.

Theorem 2.3. A mapping satisfies (1.3) if and only if there exist a symmetric multiadditive mapping , an additive mapping and a symmetric biadditive mapping such thatfor all and all .

Proof. We first assume that satisfies (1.3). Define and by for all and for all . Then, satisfies (1.1) and satisfies (1.2). By [2], there exist a symmetric multiadditive mapping and an additive mapping such thatfor all . By Lemma 2.1, there exists a symmetric biadditive mapping such thatfor all . Note that and that in general. Let be a mapping given by for all and all . Then, andfor all and all .
Since for all , and for all . Letting in (2.15), we getfor all and all . Putting in (2.15), we getfor all and all . Setting in (2.17) and using (2.16),for all and all . Taking in (2.17) and using (2.16),for all and all . By (2.18) and (2.19), for all and all . Hence, we obtain thatfor all and all .
Conversely, we assume that there exist a symmetric multiadditive mapping , an additive mapping and a symmetric biadditive mapping such thatfor all and all . Since is symmetric multiadditive and is symmetric biadditive,for all and all .

Example 2.4. Let and be the vector spaces and , respectively. Consider a function given by for all and all . One can easily verify that satisfies (1.3). By Theorem 2.3, there exist a symmetric multiadditive mapping , an additive mapping and a symmetric biadditive mapping such that for all and all . In fact,for all and all

3. Stability

From now on, let be a Banach space, and let be a function satisfyingfor all .

Lemma 3.1. Let be a mapping such that
for all . Then, there exists a unique quadratic mapping satisfying (1.2) such thatfor all . The mapping is given by for all .

Proof. Letting in (3.2), we havefor all .
Replacing by in (3.2), we get
for all . Taking in (3.2), we see thatfor all . Interchanging and in (3.6), we see thatfor all . By (3.5), (3.6), and (3.7),for all . Putting in the above inequality, we getfor all . So,wherefor all .
Thus, we obtainfor all and all . For given integers , we getfor all . By (3.13), the sequence is a Cauchy sequence for all . Since is complete, the sequence converges for all . Define byfor all . By (3.2), we havefor all and all . Letting and using (3.1), we see that satisfies (1.2). Setting and taking in (3.13), one can obtain the inequality (3.3). If is another quadratic mapping satisfying (1.2) and (3.3), we obtainfor all . Hence, is a unique quadratic mapping, as desired.

Example 3.2. If the height function in Example 2.2 satisfies for all , then there exists a unique quadratic function satisfying (1.2) such thatfor all .

Let be a function satisfying wherefor all and all . Also, let and be two functions given byfor all and all .

Theorem 3.3. Let be a mapping such that for all and all . Then, there exists a mapping satisfying (1.3) such that for all and all . The mapping is given byfor all and all .

Proof. Letting in (3.22), we havefor all and all .
Define by and for all and all . Putting in (3.22), we havefor all . Setting in (3.22), we havefor all .
By [2], there exists a cubic mapping satisfying (1.1) such thatfor all . By Lemma 3.1, there exists a quadratic mapping satisfying (1.2) such thatfor all .
If we definefor all and all , we conclude that
for all and all .

Let be a function satisfyingfor all . For the function , we can obtain similar results to Lemma 3.1 and Theorem 3.3.

As a corollary, one can obtain a result when the control mapping is a summation of terms , in which is a suitable constant by referring to [14].

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